y=x-b
y^2=2x
(x-b)^2=2x
x^2-(2b+2)x+b^2=0
Ax+Bx=2b+2
AxBx=b^2
(Ax-Bx)^2=(Ax+Bx)^2-4AxBx=8b+4
(Ay-By)^2=(Ax-Bx)^2=8b+4
|AB|=2√10
8b+4=20,b=2
Spab=|AB|h/2 h=2S/|AB|=2*2√5/(2√10)=√2
直线L平行于AB,且与AB间距h为√2
L:y=x+b' k=tana=1,cosa=√2/2
|-b-b'|cosa=h
|-b-b'|*√2/2=√2
|-b-b'|=2
b=2,b'=0或b'=-4
b'=0时y=x
y^2=2x
x^2-2x=0
x=0或x=2
Px=0,Py=0 或Px=2,Py=2
b'=-4时,y=x-4
y^2=2x
x^2-8x-2x+16=0
x^2-10X+16=0
x=2或x=8
Px=2,Py=-2或Px=8,Py=4
所以P有四解(0,0)(2,2)(2,-2)(8,4)